Carelessness?

If you want to travel through the world in a quiet, contented way, don’t get careless in any respect. Man, in every phase of life, is particularly given to carelessness. If he is on the high road to wealth and station, he becomes careless, of those who perhaps were the very means of his good fortune. On the other 'hand, if he is unfortunate in business he loses his self-respect, and rushes to the dram shop or gaming table. Whatever position a man finds himself placed in, whether by accident, fortunate speculation, or persevering industry, he should always retain that command over himself that will entitle him to the good will of old as well as new friends. If a man rises from comparative obscurity to some degree of eminence of any kindj and with no intention to offend, but carelessly notices an old friend if he meet him, he is very likely to get the ill-will of his more humble but old associate. The first time this careless recognition is noticed it produces a bad effect, and the next dis­ like, and finally hatred and contempt. We have known some of the very best friends in the world completely estranged by a wrong interpretation of acts towards each other. It is a common belief that as a man advances in the world he is desirous of cutting those who do not gain so rapidly as him­ self. This is an error, no doubt, in many instances, and the remedy is one of the easiest things in the world. A little of the starch out of the one, and the slightest liberal feeling on the other, will be found to be a true panacea for nine-tenths of the imaginary shys which lead to the entire separation of old friends, and even goes so far sometimes as to produce bad feelings among relatives. We shall end this brief article on carelessness by repeating the advice with which we begun. If you want to travel through the world in a quiet contented way, don’t get careless in any respect. Be free with your friends as though no change affected your condition in life, let that condition have changed ever so much, be it for better or worse.


Transfer errors
The most elementary errors consist of miscopying in sections of notation which do not require modification. A plus sign is copied as a minus sign, or power three becomes power two. The problem is by no means restricted to students with poor handwriting, though miniscule writing certainly does not help. Typically a student who makes a lot of errors of this kind, and who cannot find them even when told that they exist, checks his work in an unsystematic way. We can learn a lesson from proofreaders about methods of checking. One hand of a proof reader traces the original while the other hand traces the copy. The task of locating text is allocated to the hands, leaving the eyes to concentrate on checking accuracy. The text is checked in a systematic way. Using the same method students can invariably locate their copying errors, though I must admit that initially most students strongly resist the suggestion that they might use both hands to point to their work.
One type of transfer error is easy to make and strangely difficult for students to locate once made. It is the total omission of a section of notation which should be transferred without change after work has been done on an earlier section of notation. In situations where this might happen, it is helpful to encourage the students to create the shape of the answer before working out the details. Even with simple functions / and g, the derivative of the product can become f'g + g'. With more complicated functions, I encourage the students first to create the shape of the answer, ( )( ) + ( )( ), then to transfer the functions/and g to the appropriate places, ( )(g) + (/)( ), and finally to fill in the derivatives / ' and g'.
Transfer errors commonly occur on the turn of a page, particularly when algebraic manipulations approaching the limit of the student's ability for accurate work have to be performed. In this case it seems best to avoid doing any modifications to the notation on the turn over. The last line can be copied without change onto the new page. The copy can then be checked immediately against the original, perhaps by comparing the number of terms and the positions of the plus and minus signs. If transfer errors at page turning are the most elusive, then transfer errors from a question to the start of an answer must be the most insidious. There seems to be little defence against the latter type of transfer error when a student is so engrossed in the strategy of a possible answer as to recall the details of the question without reading them again. The only advice that one can give is to keep one hand on the question until work is under way, and to start any checking procedure at the question, not merely at the start of the answer.
Some simple errors in algebraic manipulation appear to be little more than transfer errors, though one needs to know the general level of work of a student to be able to judge whether such an error arose as a slip or points to a lack of understanding. For students who understand the relevant algebra, checking the accuracy of algebraic manipulation is scarcely more difficult than checking for miscopying. In this case instead of tracing the original with one finger, various parts of the original may have to be located with several fingers. The first figure reproduces part of a student's work, and shows him locating an error. A student who had learned to use simple numerical checks would not need to employ tracing techniques to locate the error in the figure. Since all the coefficients in the middle line must be divisible by 5, the last coefficient 2 must be incorrect.

Locating an error
The phenomenon of omission is seen as often in algebraic work as in copying, and there it may be even more difficult for a student to spot. It may occur when implied brackets should have been made explicit to avoid an error such as the one in the following example.

I -x
In this case it is helpful for students to write in the brackets implied in the denominator when they need to check their working. Indeed, many students who are still having difficulty with algebraic manipulation in the first year at university find that the time spent writing in brackets where conventionally they are omitted is amply repaid by the consequent improvement in accuracy.
Fortunately, similar methods are effective in the control of transfer errors, whether miscopying or omission or simple algebraic errors. If students can be persuaded to keep both hands on their work, picking out the relevant parts of the last line with one hand while writing the new line with the other, then the occurrence of transfer errors can be greatly reduced. For example, in multiplying out a bracket, a right handed writer could hold the factor outside the bracket with his left thumb while the left forefinger traced along the terms within the bracket. The method is highly effective in controlling sign and power errors.
Since transfer errors are essentially simple in nature, the methods which can be adopted to try to control their occurrence are equally simple, though I admit that getting students to adopt them is by no means simple. Nevertheless, even with those students most prone to them, the occurrence of transfer errors can be reduced significantly.

MisrecoUection
The second type of error which is commonly labelled careless involves misrecoUection of formulae which one would expect a student to use accurately. The misrecoUection may be due to the intrinsic complexity of a related set of formulae. It may be due to the need to recollect a simple formula while concentrating on other aspects of an extended problem, or to the need to apply a formula in a situation where overlying patterns of notation confuse the student.
My work with students who have a pronounced history of misrecoUection indicates that they treat mathematical facts in isolation from each other, and use half-recollected formulae without checking. The only method I know to help these students starts with the creation of diagrams of logically connected formulae and facts. The following chart is an information sheet on trigonometric functions created with a sixth form student to help him overcome a problem of frequent misrecoUection in preparation for single subject A-level mathematics. The core of this information sheet is the block which contains the basic definitions of the trigonometric functions of acute angles, presented in very condensed form. This was the information with which the student was really secure, and to which everything else could be referred for confirmation. Other blocks of information relate to this core and to each other. Blocks of information, rather than isolated facts, can be recalled as a whole; and the details of partly recalled information can be checked by reference to linked blocks of information.
For example, many of my students have difficulty with the addition formulae for sine and cosine. They try to recall the four formulae independently of each other and frequently make mistakes in doing so. They soon learn to set down the partly recalled block of information.

(A + B) + c(A -B) = 2c4cB c(/l -B) -c(A +B) = 2sAsB
A trigonometric fact-sheet plus and minus signs some students prefer to use A = 0 with sin(-B) = -sinB and cos(-B) = cosB. Others prefer to use something like A = B = jt/4. The details of the partly recalled information are checked quickly, and the one required equation is set out as a marginal note ready for use in a problem. The tasks of recollecting and checking are separated, and the details of the equations are not lost while the student is using them in a problem. Again, most of my students have seen the formulae for converting products of sines and cosines into sums or differences of sines and cosines, but have abandoned them as too involved to remember. Yet, as the chart indicates, they follow immediately from the addition formulae, and the process is simply addition or subtraction. Thus the note gives one of the equations in brief. From this description the process may seem to be very time consuming. But it becomes a quick and secure way of working. In abbreviated notation, the standard marginal note to control sign errors when differentiating and integrating sines and cosines becomes D s = c Dc = -s c = s s = -c with the signs of the derivatives checked against the graphs of sine and cosine for acute angles. The note takes far less time to write than it takes many students to worry about where the minus sign comes in integration. The frequency of misrecollections concerning exponential and logarithmic functions can also be controlled by a quick standard note which includes two valid formulae and warnings of two cases where there are no easy simplifications.

ga + b =ePgb gPb^
The last warning might be the most frequently used, since one so often sees logarithms treated as linear functions when students are using them in the middle of a difficult problem. Why is it that so many students lose their grip of elementary facts while they are working on problems? When students are under pressure all sorts of operations tend to become linear. For example, one sees log (a + b) = log a + log A and (a + b) 2 = a 2 + b 2 particularly when a and b are expressions. Students seem to lose their grip on the algebra when they are working at the two levels of recollecting a formula in terms of the letters a and b while applying the formula to a situation where a and b have to be replaced by expressions which might themselves include the letters a and b. The most effective method seems to be to encourage the students to separate the two levels of activity. A marginal note of the formula that they propose can be made in terms of the letters with which they are familiar. If there is any doubt, it can be checked for validity using simple numerical examples. Sometimes students then find it helpful to change the presentation of the formula, using brackets to emphasize its shape. For example, a marginal note such as L + L = k±± 1 , 1 _< > + ( ) a b ab ( ) < > ( ) < > helps students to separate the tasks of recalling a formula and applying it. In the application the required expressions can be filled into the brackets and parentheses. Tiredness or distraction can lead even the best students to interchange differentiation and integration, and often it occurs in the simplest situations in which the students are lulled into a false sense of familiarity. It is not uncommon to see J x 3 dx = 3x 2 in the middle of the solution to a problem. Students can be encouraged to recognise the occasions when for them it would be wise to make the marginal note Dx" = nx"~\ \x n = , « # -l . J n+1 For most students this will not be often, but for many students the marginal note f 1 .
-, 1 -= lnx, Dlnx = -J x x will help to keep them away from common errors.
Finally, how can we help students whose attempts to use formulae become confused by notation? Some students cannot simplify expressions such as a+b a-b because they cannot sort out the patterns which overlie each other. They may see a + b with a -b as the dominant pattern, and their attempts to deal with that by going to 1 (a + b)(a-b) may completely block out the pattern of sums of reciprocals. When students are encouraged to write the patterns in this algebraic expression in terms of brackets or other placeholders they find it easier to see that the move to (a -by + (a + b) (a + bKa-b} should precede the simplification of the denominator to a 2 -b 2 .